The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of d for an algebraic number α of degree d ≥ 2. This is already enough to demonstrate the existence of transcendental numbers. Thue realised that an exponent less than d would have applications to the solution of Diophantine equations and in Thue's theorem from 1909 established an exponent . Siegel's theorem improves this to an exponent about 2√d, and Dyson's theorem of 1947 has exponent about √(2d).

can have only finitely many solutions in integers p and q. If one lets α run over the whole of the set of real numbers, not just the algebraic reals, then both Roth's conclusion and Lang's hold for almost all α. So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero.[1]

The theorem is not currently effective: that is, there is no bound known on the possible values of p,q given α.[2]Davenport & Roth (1955) showed that Roth's techniques could be used to give an effective bound for the number of p/q satisfying the inequality, using a "gap" principle.[2] The fact that we do not actually know C(ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach.

The proof technique was the construction of an auxiliary function in several variables, leading to a contradiction in the presence of too many good approximations. By its nature, it was ineffective (see effective results in number theory); this is of particular interest since a major application of this type of result is to bound the number of solutions of some diophantine equations.

LeVeque generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field. Define the heightH(ξ) of an algebraic number ξ to be the maximum of the absolute values of the coefficients of its minimal polynomial. Fix κ>2. For a given algebraic number α and algebraic number field K, the equation